All Questions
16
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60
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Where will a plane intersecting a cone maximize the distance to inscribed spheres? (V. Arnold)
A cone is cut by a plane, making a closed curve. Two spheres, each inscribed into the cone, are tangent to the plane, at points $A$ and $B$ respectively.
Find the point $C$ on the cut line (i.e. the ...
- 4,450
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votes
0
answers
63
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closed curve mapping problem
I have this interesting, seemingly difficult problem I came across while think about rendering. It seems simple enough that I suspect it might have a name already (and hopefully be solved), but I can'...
- 93
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97
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What is the connection between optimization and symmetry?
I have found that in many optimization problems, there is some sort of "local symmetry" phenomena going on. For example, suppose we have a 'nice' function which has $f'(x)=0$, then we will ...
1
vote
1
answer
164
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constrained differential equation
I'm struggeling with the derivation of the ode equations of forward kinematics of an oriented object. Assuming to be in $R^2$ and using the coordinates $(x_1,x_2,x_3):=(\phi,p_1,p_2)$, where the angle ...
- 165
1
vote
1
answer
138
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How to find geodesics in metric spaces
Let $(X,d)$ be a metric space. Let $x,y\in X$.
How do we generally formulate the problem of finding the shortest path $\gamma :[0,1] \rightarrow X
$ between $x,y$ ? Is it something like
$$\inf\...
- 2,294
3
votes
3
answers
180
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Existence of "checkerboard" grid whose vertices intersect all given points on plane
This is a question about identifying a theorem or maybe even a subject matter. I've asked myself a question, but I don't know where to look for answers. Typing a problem statement in a search engine ...
- 97.7k
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How would you go about proving that the right cone is the most optimal pyramid
Im trying to figure this out. Like you could go one by one optimizing a square pyramid by minimizing its surface area for a given volume. Eventually when you get up to an decagon based right pyramid, ...
- 3,676
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2
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123
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Can we measure total "closeness" of two geometric objects with analysis?
Say I have one sphere and one plane
$$x^2+y^2+z^2 = 1$$
$$z = 2$$
We can easily calculate closest distance between these. It is easy exercise. Maybe first exercise in differential geometry or ...
- 66.8k
1
vote
2
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77
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Must any shortest line between two surfaces coincide with normals to both surfaces?
In differential geometry I know of a result which says something along the lines of
...
- 5,507
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54
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Identify a point in the disc that is as far away from each point of S as possible Ask Question
I have a question.The question is this.
Say you have a finite set S of n points in a circular disc. Think of the points in S as houses on a circular island. You want to locate a garbage incineration ...
- 4,243
3
votes
1
answer
345
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Does this surface minimize maximum distance?
Suppose we have a tetrahedron defined by points $(0,0,0),(1,1,0),(0,1,1),(1,0,1)$. Now define surface by $(a,b,a + b - 2ab)$ for $a,b$ between $0$ and $1$. Let $E_1$ be the set of points inside the ...
- 3,965
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145
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How to compute the geometric center of a curved manifold defined by a set of point?
Suppose that I have a set of points E which belong to a manifold. Typically a surface mesh like that.
$$E=\{x_{0},x_1,...,x_n\}$$ where the $x_i$ are the points coordinates in 3D such that : $x_i = (...
- 135
6
votes
1
answer
150
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Minimizing the size of a cylinder sliding in a tube
Lets define a spine curve $\Gamma$ given in terms of its arc length parameter $s$, and a family of closed curves $\Omega = \Omega \left( s \right)$. Lets say $\Gamma$ has a finite length, from $s=0$ ...
- 35.4k
16
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2k
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Properties of the projection onto a nonconvex set
Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping
$$
g:\...
- 3,146
6
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141
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Do balls optimize the boundary area for a fixed volume?
I recently saw this question, asking if the circle is the planar figure which has the least perimeter given the area. As far as I know, this is a classical problem, and the answer is affirmative. I am ...
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